A Basic Course in Number Theory
A Basic Course in Number Theory. Instructor: Prof. Dipendra Prasad, Department of Mathematics, IIT Bombay. This course intends to develop the basics of number theory touching upon many essential points such as the prime number theorem, quadratic reciprocity laws, Gauss theorem on the classification of binary quadratic forms, Brahmagupta-Pell equations, to quote a few. This course will enable a student to learn more advanced topics in number theory.
(from nptel.ac.in
Lecture 37 - The Jacobi Symbol
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Lecture 01 - Integers
Lecture 02 - Divisibility and Primes
Lecture 03 - Infinitude of Primes
Lecture 04 - Division Algorithm and the GCD
Lecture 05 - Computing the GCD and Euclid's Lemma
Lecture 06 - Fundamental Theorem of Arithmetic
Lecture 07 - Stories around Primes
Lecture 08 - Winding up on Primes and Introducing Congruences
Lecture 09 - Basic Results in Congruences
Lecture 10 - Residue Classes Modulo N
Lecture 11 - Arithmetic Modulo N, Theory and Examples
Lecture 12 - Arithmetic Modulo N, More Examples
Lecture 13 - Solving Linear Polynomials Modulo N, Part I
Lecture 14 - Solving Linear Polynomials Modulo N, Part II
Lecture 15 - Solving Linear Polynomials Modulo N, Part III
Lecture 16 - Solving Linear Polynomials Modulo N, Part IV
Lecture 17 - Chinese Remainder Theorem, The Initial Cases
Lecture 18 - Chinese Remainder Theorem, The General Case and Examples
Lecture 19 - Chinese Remainder Theorem, More Examples
Lecture 20 - Using the CRT, Square Roots of 1 in Zn
Lecture 21 - Wilson's Theorem
Lecture 22 - Roots of Polynomials of Zp
Lecture 23 - Euler φ-Function, Part I
Lecture 24 - Euler φ-Function, Part II
Lecture 25 - Primitive Roots, Part I
Lecture 26 - Primitive Roots, Part II
Lecture 27 - Primitive Roots, Part III
Lecture 28 - Primitive Roots, Part IV
Lecture 29 - Structure of Un, Part I
Lecture 30 - Structure of Un, Part II
Lecture 31 - Quadratic Residues
Lecture 32 - The Legendre Symbol
Lecture 33 - Quadratic Reciprocity Law, Part I
Lecture 34 - Quadratic Reciprocity Law, Part II
Lecture 35 - Quadratic Reciprocity Law, Part III
Lecture 36 - Quadratic Reciprocity Law, Part IV
Lecture 37 - The Jacobi Symbol
Lecture 38 - Binary Quadratic Forms
Lecture 39 - Equivalence of Binary Quadratic Forms
Lecture 40 - Discriminant of a Binary Quadratic Form
Lecture 41 - Reduction Theory of Integral Binary Quadratic Forms
Lecture 42 - Reduced Forms up to Equivalence, Part I
Lecture 43 - Reduced Forms up to Equivalence, Part II
Lecture 44 - Reduced Forms up to Equivalence, Part III
Lecture 45 - Sums of Squares, Part I
Lecture 46 - Sums of Squares, Part II
Lecture 47 - Sums of Squares, Part III
Lecture 48 - Beyond Sums of Squares, Part I
Lecture 49 - Beyond Sums of Squares, Part II
Lecture 50 - Continued Fractions - Basic Results
Lecture 51 - Dirichlet's Approximation Theorem
Lecture 52 - Good Rational Approximations
Lecture 53 - Continued Fraction Expansion for Real Numbers, Part I
Lecture 54 - Continued Fraction Expansion for Real Numbers, Part II
Lecture 55 - Convergents Give Better Approximations
Lecture 56 - Convergents are the Best Approximations, Part I
Lecture 57 - Convergents are the Best Approximations, Part II
Lecture 58 - Quadratic Irrationals as Continued Fractions
Lecture 59 - Some Basics of Algebraic Number Theory
Lecture 60 - Units in Quadratic Fields: The Imaginary Case
Lecture 61 - Units in Quadratic Fields: The Real Case
Lecture 62 - Brahmagupta-Pell Equations
Lecture 63 - Tying Some Loose Ends